Abstract

Photoluminescence is one of the processes by which photons are emitted after the absorption of incoming photons at a higher energy. But the yield and spectral band shape of the emission can be altered by the optical properties of the luminophore environment through scattering and absorption. To understand these effects on a photoluminescent turbid layer, the Kubelka–Munk model, which is a two-flux approximation of the radiative transfer equation, can be used. Compared to previous works, this translucent layer can be applied on a colored opaque background. The model takes into account the absorption, scattering, and luminescent properties of the layer and the reflection by the background, for both the light excitation and the light emission. The competition between these different optical interactions is studied; e.g., the model can predict the presence of an emission maximum by increasing the thickness of the luminescent layer on a light background. Moreover, the model is extended to two important cases: the presence of a photoluminescent background and the effect of a refractive index discontinuity.

© 2011 Optical Society of America

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  1. M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
    [CrossRef]
  2. R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
    [CrossRef]
  3. J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
    [CrossRef] [PubMed]
  4. S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
    [CrossRef]
  5. E. Allen, “Fluorescent white dyes: calculation of fluorescence from reflectivity values,” J. Opt. Soc. Am. 54, 506–515 (1964).
    [CrossRef]
  6. R. G. Hersch, “Spectral prediction model for color prints on paper with fluorescent additives,” Appl. Opt. 47, 6710–6722 (2008).
    [CrossRef] [PubMed]
  7. T. H. Morton, “Fluorescent brightening agents on textiles: elementary optical theory and its practical applications,” J. Soc. Dyers Colour. 79, 238–242 (2008).
    [CrossRef]
  8. E. R. De la Rie, “Fluorescence of paint and varnish layers (part II),” Stud. Conserv. 27, 65–69 (1982).
    [CrossRef]
  9. T. Miyoshi, “Fluorescence from oil colours, linseed oil and poppy oil under N//2 laser excitation,” Jpn. J. Appl. Phys. 24, 371–372 (1985).
    [CrossRef]
  10. M. Thoury, M. Elias, J. M. Frigerio, and C. Barthou, “Nondestructive varnish identification by ultraviolet fluorescence spectroscopy,” Appl. Spectrosc. 61, 1275–1282 (2007).
    [CrossRef]
  11. G. Verri, C. Clementi, D. Comelli, S. Cather, and F. Piqué, “Correction of ultraviolet-induced fluorescence spectra for the examination of polychromy,” Appl. Spectrosc. 62, 1295–1302 (2008).
    [CrossRef] [PubMed]
  12. C. Clementi, C. Miliani, G. Verri, S. Sotiropoulou, A. Romani, B. G. Brunetti, and A. Sgamellotti, “Application of the Kubelka–Munk correction for self-absorption of fluorescence emission in carmine lake paint layers,” Appl. Spectrosc. 63, 1323–1330(2009).
    [CrossRef] [PubMed]
  13. M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
    [CrossRef]
  14. J. K. Delaney, J. G. Zeibel, M. Thoury, R. Littleton, M. Palmer, K. M. Morales, E. R. De La Rie, and A. Hoenigswald, “Visible and infrared imaging spectroscopy of Picasso’s Harlequin Musician: mapping and identification of artist materials in situ,” Appl. Spectrosc. 64, 584–594 (2010).
    [CrossRef] [PubMed]
  15. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).
  16. G. Kortüm, Reflectance Spectroscopy (Springer Verlag, 1969), pp. 103–123 and 152–156.
  17. L. Fukshansky and N. Kazarinova, “Extension of the Kubelka–Munk theory of light propagation in intensively scattering materials to fluorescent media,” J. Opt. Soc. Am. 70, 1101–1111(1980).
    [CrossRef]
  18. A. Kokhanovsky, “Radiative properties of optically thick fluorescent turbid media,” J. Opt. Soc. Am. A 26, 1896–1900 (2009).
    [CrossRef]
  19. A. Kokhanovsky, “Erratum: radiative properties of optically thick fluorescent turbid media,” J. Opt. Soc. Am. A 27, 2084–2084 (2010).
    [CrossRef]
  20. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
    [CrossRef]
  21. D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).
  22. L. Simonot, “A photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
    [CrossRef] [PubMed]

2010 (3)

2009 (5)

A. Kokhanovsky, “Radiative properties of optically thick fluorescent turbid media,” J. Opt. Soc. Am. A 26, 1896–1900 (2009).
[CrossRef]

L. Simonot, “A photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009).
[CrossRef] [PubMed]

C. Clementi, C. Miliani, G. Verri, S. Sotiropoulou, A. Romani, B. G. Brunetti, and A. Sgamellotti, “Application of the Kubelka–Munk correction for self-absorption of fluorescence emission in carmine lake paint layers,” Appl. Spectrosc. 63, 1323–1330(2009).
[CrossRef] [PubMed]

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

2008 (3)

2007 (1)

1998 (1)

M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
[CrossRef]

1993 (1)

1985 (1)

T. Miyoshi, “Fluorescence from oil colours, linseed oil and poppy oil under N//2 laser excitation,” Jpn. J. Appl. Phys. 24, 371–372 (1985).
[CrossRef]

1982 (1)

E. R. De la Rie, “Fluorescence of paint and varnish layers (part II),” Stud. Conserv. 27, 65–69 (1982).
[CrossRef]

1980 (1)

1964 (1)

1942 (2)

L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
[CrossRef]

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).

1931 (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Allen, E.

Barthou, C.

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

M. Thoury, M. Elias, J. M. Frigerio, and C. Barthou, “Nondestructive varnish identification by ultraviolet fluorescence spectroscopy,” Appl. Spectrosc. 61, 1275–1282 (2007).
[CrossRef]

Brunetti, B. G.

Cather, S.

Clementi, C.

Comelli, D.

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

G. Verri, C. Clementi, D. Comelli, S. Cather, and F. Piqué, “Correction of ultraviolet-induced fluorescence spectra for the examination of polychromy,” Appl. Spectrosc. 62, 1295–1302 (2008).
[CrossRef] [PubMed]

Davis, S. C.

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

De La Rie, E. R.

Dehghani, H.

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

Delaney, J. K.

Dicelio, L. E.

M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
[CrossRef]

Elias, M.

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

M. Thoury, M. Elias, J. M. Frigerio, and C. Barthou, “Nondestructive varnish identification by ultraviolet fluorescence spectroscopy,” Appl. Spectrosc. 61, 1275–1282 (2007).
[CrossRef]

Feld, M. S.

Frigerio, J. M.

Fukshansky, L.

Goulas, Y.

R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
[CrossRef]

Hersch, R. G.

Hoenigswald, A.

Jacquemoud, S.

R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
[CrossRef]

Judd, D. B.

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).

Kazarinova, N.

Kokhanovsky, A.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer Verlag, 1969), pp. 103–123 and 152–156.

Kubelka, P.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Lagorio, M. G.

M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
[CrossRef]

Litter, M.

M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
[CrossRef]

Littleton, R.

Louis, J.

R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
[CrossRef]

Magnain, C.

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

Miliani, C.

Miyoshi, T.

T. Miyoshi, “Fluorescence from oil colours, linseed oil and poppy oil under N//2 laser excitation,” Jpn. J. Appl. Phys. 24, 371–372 (1985).
[CrossRef]

Morales, K. M.

Morton, T. H.

T. H. Morton, “Fluorescent brightening agents on textiles: elementary optical theory and its practical applications,” J. Soc. Dyers Colour. 79, 238–242 (2008).
[CrossRef]

Moya, I.

R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
[CrossRef]

Munk, F.

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Nevin, A.

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

Palmer, M.

Paulsen, K. D.

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

Pedrós, R.

R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
[CrossRef]

Piqué, F.

Pogue, B. W.

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

Rava, R. P.

Roman, E. San

M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
[CrossRef]

Romani, A.

Saunderson, L.

Sgamellotti, A.

Simonot, L.

Sotiropoulou, S.

Thoury, M.

Tuttle, S. B.

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

Valentini, G.

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

Verri, G.

Wu, J.

Zeibel, J. G.

Appl. Opt. (3)

Appl. Spectrosc. (4)

J. Appl. Phys. (1)

S. C. Davis, B. W. Pogue, S. B. Tuttle, H. Dehghani, and K. D. Paulsen, “Spectral distortion in diffuse molecular luminescence tomography in turbid media,” J. Appl. Phys. 105, 102024 (2009).
[CrossRef]

J. Chem. Soc. Faraday Trans. (1)

M. G. Lagorio, L. E. Dicelio, M. Litter, and E. San Roman, “Modeling of fluorescence quantum yields of supported dyes,” J. Chem. Soc. Faraday Trans. 94, 419–425 (1998).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Res. Natl. Bur. Stand. (1)

D. B. Judd, “Fresnel reflection of diffusely incident light,” J. Res. Natl. Bur. Stand. 29, 329–332 (1942).

J. Soc. Dyers Colour. (1)

T. H. Morton, “Fluorescent brightening agents on textiles: elementary optical theory and its practical applications,” J. Soc. Dyers Colour. 79, 238–242 (2008).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Miyoshi, “Fluorescence from oil colours, linseed oil and poppy oil under N//2 laser excitation,” Jpn. J. Appl. Phys. 24, 371–372 (1985).
[CrossRef]

Proc. SPIE (1)

M. Elias, C. Magnain, C. Barthou, A. Nevin, D. Comelli, and G. Valentini, “UV-fluorescence spectroscopy for identification of varnishes in works of art: influence of the underlayer on the emission spectrum,” Proc. SPIE 7391, 739104 (2009).
[CrossRef]

Remote Sens. Environ. (1)

R. Pedrós, Y. Goulas, S. Jacquemoud, J. Louis, and I. Moya, “FluorMODleaf: a new leaf fluorescence emission model based on the PROSPECT model,” Remote Sens. Environ. 114, 155–167 (2010).
[CrossRef]

Stud. Conserv. (1)

E. R. De la Rie, “Fluorescence of paint and varnish layers (part II),” Stud. Conserv. 27, 65–69 (1982).
[CrossRef]

Z. Tech. Phys. (1)

P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593–601 (1931).

Other (1)

G. Kortüm, Reflectance Spectroscopy (Springer Verlag, 1969), pp. 103–123 and 152–156.

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Figures (8)

Fig. 1
Fig. 1

Schematic description of the system.

Fig. 2
Fig. 2

Influence of the background reflectance with K E = 0.1 mm 1 and S E = 0.1 mm 1 and R g A = R g E . (a) Normalized emission flux F m / ( ϕ F I 0 ) plotted against the thickness with K A = 1 mm 1 and S A = 1 mm 1 . (b) Reflectance plotted against the thickness.

Fig. 3
Fig. 3

Influence of the absorption coefficient K E with S E = 0.1 mm 1 and R g A = R g E = 1 (white background). (a) Normalized emission flux F m / ( ϕ F I 0 ) plotted against the thickness with K A = 1 mm 1 and S A = 1 mm 1 . (b) Reflectance plotted against the thickness.

Fig. 4
Fig. 4

Influence of the backscattering coefficient S E with K E = 0.1 mm 1 and R g A = R g E = 1 (white background). (a) Normalized emission flux F m / ( ϕ F I 0 ) plotted against the thickness with K A = 1 mm 1 and S A = 1 mm 1 . (b) Reflectance plotted against the thickness.

Fig. 5
Fig. 5

Reflection (R), transmission (T), and total ( R + T ) modes for a single layer with K E = 0.1 mm 1 and S E = 0.1 mm 1 . (a) Normalized emission flux F m / ( ϕ F I 0 ) plotted against the thickness with K A = 1 mm 1 and S A = 1 mm 1 . (b) Reflectance and transmittance plotted against the thickness.

Fig. 6
Fig. 6

Influence of the background reflectance for a nonscattering layer ( S A = S E = 0 mm 1 ) with K A = 1 mm 1 , K E = 0.1 mm 1 , and R g A = R g E . (a) Normalized emission flux F m / ( ϕ F I 0 ) plotted against the thickness. (b) Reflectance plotted against the thickness.

Fig. 7
Fig. 7

Nonluminescent layer on a luminescent background. Evolution of the emission flux normalized by the emission without the layer in term of the layer thickness with K A = 1 mm 1 , S A = 1 mm 1 , K E = 0.1 mm 1 , S E = 0.1 mm 1 , and R g A = R g E .

Fig. 8
Fig. 8

Variation of the ratio F m c / F m [Eq. (46)] in terms of the refractive index ratio n 1 / n 0 for different background reflectance R g A = R g E . The refractive indices are assumed to be real and wavelength independent.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

α = K ( K + 2 S ) ,
β = K / ( K + 2 S ) ,
{ B 1 = 1 + β R g ( 1 β ) B 2 = 1 β R g ( 1 + β ) .
R = 1 β 1 + β .
K S = ( 1 R ) 2 2 R .
I A ( z ) = ( K A + S A ) I A ( z ) + S A J A ( z ) ,
J A ( z ) = ( K A + S A ) J A ( z ) + S A I A ( z ) ,
I E ( z ) = ( K E + S E ) I E ( z ) + S E J E ( z ) + P ( z ) ,
J E ( z ) = ( K E + S E ) J E ( z ) + S E I E ( z ) + P ( z ) .
P ( z ) = 1 / 2 ϕ F K A [ I A ( z ) + J A ( z ) ] .
I A ( z = 0 ) = I 0 ,
J A ( z = d ) = R g A I A ( z = d ) ,
I E ( z = 0 ) = 0 ,
J E ( z = d ) = R g E I E ( z = d ) ,
{ I A ( z ) = A 1 ( 1 β A ) e α A z + A 2 ( 1 + β A ) e α A z J A ( z ) = A 1 ( 1 + β A ) e α A z + A 2 ( 1 β A ) e α A z ,
{ A 1 = B 2 A e α A d ( 1 + β A ) B 1 A e α A d ( 1 β A ) B 2 A e α A d I 0 A 2 = B 1 A e α A d ( 1 + β A ) B 1 A e α A d ( 1 β A ) B 2 A e α A d I 0 .
P ( z ) = ϕ F K A ( A 1 e α A z + A 2 e α A z ) .
J E ( z ) α E 2 J E ( z ) = Φ ( z ) ,
Φ ( z ) = ( α E / β E ) P ( z ) P ( z ) .
Φ ( z ) = C 1 e α A z + C 2 e α A z ,
{ C 1 = ϕ F K A A 1 ( α E / β E + α A ) C 2 = ϕ F K A A 2 ( α E / β E α A ) .
J H ( z ) = C 3 ( 1 + β E ) e α E z + C 4 ( 1 β E ) e α E z ,
J p ( z ) = C 1 α A 2 α E 2 e α A z + C 2 α A 2 α E 2 e α A z .
J E ( z ) = C 1 α A 2 α E 2 e α A z + C 2 α A 2 α E 2 e α A z + C 3 ( 1 + β E ) e α E z + C 4 ( 1 β E ) e α E z .
I E ( z ) = D 1 α A 2 α E 2 e α A z + D 2 α A 2 α E 2 e α A z + C 3 ( 1 β E ) e α E z + C 4 ( 1 + β E ) e α E z ,
{ D 1 = ϕ F K A A 1 ( α E / β E α A ) D 2 = ϕ F K A A 2 ( α E / β E + α A ) .
{ C 3 = 1 α A 2 α E 2 B 2 E ( D 1 + D 2 ) e α E d ( 1 + β E ) ( ( C 1 R g E D 1 ) e α A d + ( C 2 R g E D 2 ) e α A d ) ( 1 + β E ) B 1 E e α E d ( 1 β E ) B 2 E e α E d C 4 = 1 1 + β E ( D 1 + D 2 α A 2 α E 2 + C 3 ( 1 β E ) ) .
F m = J E ( z = 0 ) = C 1 α A 2 α E 2 + C 2 α A 2 α E 2 + C 3 ( 1 + β E ) + C 4 ( 1 β E ) .
J A ( z = 0 ) = A 1 ( 1 + β A ) + A 2 ( 1 β A ) .
A 1 = C 1 = D 1 = C 3 = 0 ,
F m = C 2 α A 2 α E 2 + C 4 ( 1 β E ) ,
{ A 2 = I 0 1 + β A C 2 = ϕ F K A A 2 ( α E / β E α A ) D 2 = ϕ F K A A 2 ( α E / β E + α A ) C 4 = 1 1 + β E ( D 2 α A 2 α E 2 ) .
F m = 1 2 ϕ F I 0 K A 1 α A + α E 2 1 + β A 2 1 + β E .
F m = 1 2 ϕ F I 0 K A ( 1 + R A ) ( 1 + R E ) α A + α E .
F m = 1 2 ϕ F I 0 ( K A S A ) ( 1 + R A ) ( 1 + R E ) b A + b E ,
b = K S ( K S + 2 ) .
F = ϕ F I 0 ( 1 R A ) .
{ B 1 = 1 + β B 2 = 1 β C 3 = 1 α A 2 α E 2 B 2 E ( D 1 + D 2 ) e α E d ( 1 + β E ) ( C 1 e α A d + C 2 e α A d ) ( 1 + β E ) B 1 E e α E d ( 1 β E ) B 2 E e α E d .
I E ( z = d ) = D 1 α A 2 α E 2 e α A d + D 2 α A 2 α E 2 e α A d + C 3 ( 1 β E ) e α E d + C 4 ( 1 + β E ) e α E d .
F m = 1 2 ϕ F I 0 K A [ R g E e 2 K E d R g A e 2 K A d + ( R g A R g E ) e ( K A + K E ) d K A K E + 1 R g A R g E e 2 ( K A + K E ) d + ( R g A R g E 1 ) e ( K A + K E ) d K A + K E ] .
J E ( z = d ) = R g E I E ( z = d ) + F g I A ( z = d ) .
I A ( z = d ) = A 1 ( 1 β A ) e α A d + A 2 ( 1 + β A ) e α A d .
J E ( z = 0 ) = F m + 4 β E ( 1 + β E ) B 1 E e α E d ( 1 β E ) B 2 E e α E d F g I A ( z = d ) ,
{ I 1 = t 01 I 0 + r 10 J 1 J 0 = r 01 I 0 + t 10 J 1 ,
R c = r 01 + t 01 t 10 R 1 r 10 R .
F m c = t 01 A 1 r 10 A R A t 10 E 1 r 10 E R E F m .

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